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Theorem nfald 1760
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfald (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . . 4 𝑦𝜑
21nfri 1519 . . 3 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
42, 3alrimih 1469 . 2 (𝜑 → ∀𝑦𝑥𝜓)
5 nfnf1 1544 . . . 4 𝑥𝑥𝜓
65nfal 1576 . . 3 𝑥𝑦𝑥𝜓
7 hba1 1540 . . . 4 (∀𝑦𝑥𝜓 → ∀𝑦𝑦𝑥𝜓)
8 sp 1511 . . . . 5 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
98nfrd 1520 . . . 4 (∀𝑦𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
107, 9hbald 1491 . . 3 (∀𝑦𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
116, 10nfd 1523 . 2 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝑦𝜓)
124, 11syl 14 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wnf 1460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  dvelimALT  2010  dvelimfv  2011  nfeudv  2041  nfeqd  2334  nfraldw  2509  nfraldxy  2510  nfiotadw  5176  nfixpxy  6710  bdsepnft  14261
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